Denote by \({\rm Sp}(2n)\)  the symplectic group of real \(2n\times 2n\)  matrices.
Following [1]}, for any \(\omega \in {\bf U}=\lbrace z\in {\bf C}\;|\;|z|=1\rbrace \) , define a real function
\(D_{\omega }(M)=(-1)^{n-1}\overline{{\omega }}^n det(M-{\omega }I_{2n})\)  for any \(M\)  in the symplectic group \({\rm Sp}(2n)\) .
Then we can define \({\rm Sp}(2n)_{{\omega }}^0 = \lbrace M\in {\rm Sp}(2n)\,|\, D_{{\omega }}(M)=0\rbrace \)  and
\({\rm Sp}(2n)_{{\omega }}^{\ast } = {\rm Sp}(2n)\setminus {\rm Sp}(2n)_{{\omega }}^0\) . The orientation of \({\rm Sp}(2n)_{{\omega }}^0\)  at any of its point
\(M\)  is defined to be the positive direction \(\frac{{\mathrm {d}}}{{\mathrm {d}}t}Me^{t J}|_{t=0}\)  of the path \(Me^{t J}\)  with \(t>0\)  small
enough. Let \(\nu _{{\omega }}(M)=\dim _{{\bf C}}\ker _{{\bf C}}(M-{\omega }I_{2n})\) . Let
\(\mathcal {P}_{2\pi }(2n) = \lbrace {\gamma }\in C([0,2\pi ],{\rm Sp}(2n))\;|\;{\gamma }(0)=I\rbrace \)  and
\({\gamma }_0(t)={\rm diag}(2-\frac{t}{2\pi }, (2-\frac{t}{2\pi })^{-1})\)  for \(0\le t\le 2\pi \) .
